Lemma 1 (Lemma 1 in [1]}) 
Suppose the probability distribution of the \(n_N\)  random variables are multi-variant Gaussian, and \(\sup _{i, j \in \lbrace 1,\ldots , n_N\rbrace }|\rho _{ij}| \le M < 1\) , then
\(\sup _{(i,j, \mathcal {T}_{i,j}^m) \cup (i, j, \mathcal {T}_{i,j})} P(|r_{ij} - \rho _{ij}| > \gamma ) \le C_1(M) \cdot (N-2) \cdot \exp \Big ( (N-4)\log \frac{4-\gamma ^2}{4+\gamma ^2} \Big ),\) 
